1,720,985 research outputs found
When the Morse index is infinite
No full textWe prove that if f is a functional on a Hilbert manifold M having critical points with infinite Morse index and co-index, the following fact holds: for every arbitrary choice of an integer a(x) for each critical point x, there exists a Riemannian metric on M such that the gradient flow of f is Morse-Smale and the intersection of the unstable manifold of x with the stable manifold of y has dimension a(x)-a(y). This fact shows that for strongly indefinite functionals, no Morse theory based only on the data (M,f) can exist
On the global stable manifold
Full text linkWe give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the global stable manifold, a problem which in the general Banach setting gives rise to subtle questions about the possibility of extending germs of diffeomorphisms
A Morse complex for Lorentzian geodesics
No full textWe prove the Morse relations for the set of all geodesics connecting two non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We overcome the difficulties coming from the fact that the Morse index of every geodesic is infinite, and from the lack of the Palais-Smale condition, by using the Morse complex approach
Infinite dimensional Grassmannians
No full textWe study the differentiable structure and the homotopy type of some spaces related to the Grassmannian of closed linear subspaces of an infinite dimensional Hilbert space, such as the space of Fredholm pairs, the Grassmannian of compact perturbations of a given space, and the essential Grassmannians. We define a determinant bundle over the space of Fredholm pairs. We lift the composition of Fredholm operators to the Quillen determinant bundle, and we show how this map can be used in several constructions involving the determinant bundle over the space of Fredholm pairs. We deduce some properties of suitable orientation bundles
Gap phenomenon for autonomous functionals
No full textWe give an example of an autonomous functional (where is open subset of and belongs the Sobolev space ) which is sequentially weakly lower
semicontinuous in for every but does not agree with the relaxation of the same functional restricted to smooth functions when .
A Lavrentiev phenomenon occurs for a related boundary problem
A non-squeezing theorem for convex symplectic images of the Hilbert ball
No full textWe prove that the non-squeezing theorem of Gromov holds for symplectomor- phisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by dual- ity methods of a symplectic capacity for bounded convex neighbourhoods of the ori- gin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions
The Morse complex for infinite dimensional manifolds, Part I
No full textInthispaperandintheforthcomingPartII,weintroduceaMorsecomplexforaclass of functions f defined on an infinite dimensional Hilbert manifold M, possibly having critical pointsofinfiniteMorseindexandco-index.Theideaistoconsideraninfinitedimensional subbundle—or more generally an essential subbundle—of the tangent bundle of M, suitably related with the gradient flow of f. This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y: finite dimensionality of W, possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W, classification, up to topological conjugacy, of the gradient flow on the closure of W, in the case dim W =2
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